Recently, a new formalism for studying the thermodynamics of black holes has been proposed by including the parameters central charge and the chemical potential as a new couple of conjugate thermodynamics variables [1]}. Also, in the dual theory, the square number of colors is connected to the central charge [2]}, [3]}, [4]}, [5]}. Here, we focus on the Taub-NUT-AdS black hole in the restricted phase space (RPS) and study the thermodynamic properties of the model in the presence of the central charge. As the key point of the present work, the weak gravity conjecture (WGC) for such a model can be easily proved at the critical points.
